FROM BALLISTIC TO DIFFUSIVE REGIMES IN HEAT TRANSPORT
AT NANO SCALES
Georgy Lebon1, Miroslav Grmela2, Charles Dubois2.
1Université
de Liège, Dpt d’Astrophysique, Géophysique et Océanographie, 17, allée du 6 Août, 4000 Liège, Belgique
Polytechnique, Center for Applied Research on Polymers and Composites, Montreal, Canada
2Ecole
Contact email address: g.lebon@ulg.ac.be
ABSTRACT
Our purpose is to model heat transport at micro and nano-scales. At these scales, the mechanism of transport is not only diffusive,
i.e. dominated by multiple scattering of phonons, but includes also a ballistic contribution due to collisions of phonons with the
boundaries. To takes these effects into account, we follow the line of thoughts of Extended Irreversible Thermodynamics. The
original idea underlying the present paper is to assume the co-existence of two kinds of heat carriers, namely ballistic and
diffusive phonons, which are obeying different time-evolution equations: the diffusive collisions are described by a Cattaneo
equation and ballistic interactions by a Guyer-Krumhansl like-equation. The model is applied to the problem of transient heat
transport in micro films and the results are compared with earlier descriptions.
KEYWORDS: Diffusion, heat transport, irreversible thermodynamics, nano systems.
1. INTRODUCTION
Micro- and nano-scale systems have known an extraordinary fast development during the last decade due to
the advancement of thin film technology and nano-devices. At these small sizes, the usual laws of
macroscopic continuum physics should be revisited to take into account of the boundary and interface
effects as well as the finite relaxation time of energy carriers. In particular, it has been recognized that the
Fourier law of heat conduction breaks down at nano-scales: it is only applicable at large wave-length and
large time scales. Indeed, it was noticed, some decades ago, that Fourier's law is not applicable to high
frequency processes as it predicts propagation of signals with an infinite velocity. Recently, G. Chen [1] has
proposed an interesting model, the so-called ballistic-diffusion equation, in order to describe transient heat
conduction across thin films. Chen's approach, based on Boltzmann’s equation, consists in decomposing the
distribution function in two parts: a diffusive and a ballistic one. As a consequence, the total (specific)
internal energy u will be decomposed into a diffusive um and a ballistic contribution ub in such a way that u
= ud + ub. Similarly the heat flux vector q will be split into a diffusive qd and a ballistic qb contribution so
that q = q d +q b.
Our objective is to show that Chen's results can be described and recovered within the framework of
Extended Irreversible Thermodynamics [2,3 ]. The present study complements Chen's approach which is
essentially based on Boltzmann's equation and the kinetic theory. It allows also to enlighten some results
derived by Chen and may open interesting perspectives for future developments. The paper will run as
follows. The main deficiencies of the classical Fourier and Cattaneo [4] equations as welle as the
ingredients of Irreversible Thermodynamics are briefly recalled in section 2. In the next section, a
thermodynamic model of heat conduction at nano-scales, involving the coexistence of ballistic and diffusive
heat carriers, is proposed. The description is applied in Section 4 to the study of transient heat conduction in
thin micro films; the results are compared with these provided by other descriptions. Final comments and
conclusion are drawn in section 5.
2. DIFFUSIVE AND BALLISTIC HEAT CARRIERS
For isotropic systems, heat conduction is well described by the classical Fourier law
q  T ,
(1)
which relates linearly the heat flux vector q to the gradient of temperature T, λ. designating the heat
conductivity. This relation implies an instantaneous response of the system and is therefore no longer valid
when the perturbation is very fast or when the response time is very long. This happens, in particular, in
heat conduction in miniaturized systems, such as nano-wires and thin layers submitted to high-frequency
perturbations. Heat transport in such systems differs significantly from the predictions of Fourier’s law,
because their behaviour is strongly influenced by memory and nonlocal effects Moreover, at high
frequencies, one has also to introduce the relaxation time of the heat flux to take into account of the finite
response time to perturbations. This has motivated the introduction of a more general heat transport
equation referred to as Cattaneo’s equation [4]
 q& q  T ,
(2)
wherein τ denotes a suitable relaxation time, while a superposed dot stands for the material time derivative
which, in the case of rigid bodies at rest considered in the present work, is identified as the partial time
derivative. When τ is negligible, or when the time variation of the heat flux is slow, Eq. (2) will reduce to
Fourier law. It is worth to stress that Cattaneo’s equation (2) is not the most general one to describe heat
conduction in miniaturized systems. For instance, Chen [1] regarded the transport of heat in nanosize
devices as a combination of ballistic phonons described by Boltzmann’s kinetic equation and diffusive
phonons obeying Cattaneo’s Eq. (2). In the present work, we adopt Chen’s point of view assuming the
coexistence of ballistic and the diffusive heat carriers. However, in contrast with Chen, we do not refer to
the kinetic Boltzmann equation but assume that ballistic phonons satisfy a time-evolution equation
incorporating non-local effect. Such an attitude is typical of Extended Irreversible Thermodynamics (EIT)
wherein the dissipative fluxes are elevated to the status of independent variables at the same footing as the
classical variables as temperature, or energy.
3. STATE VARIABLES AND TIME-EVOLUTION EQUATIONS
In nano- and micro-systems, the ratio between the mean free path of the heat carriers (phonons) and the
characteristic length of the system given by the Knudsen number becomes comparable or larger than one.
The mechanism of heat transfer is no longer of diffusive nature, i.e. dominated by multiple scattering
between the carriers but it includes ballistic contributions resulting essentially from collisions with the
boundaries. To cope with the ballistic effects, we shall with Chen [1] decompose the specific internal
energy u and the heat flux vector q into a diffusive and a ballistic contributions, and select the following
set of state variables:
i) the couple ud , qd to account for the diffusive behaviour of the heat carriers;
ii) the couple ub,, qb to provide a description of the ballistic microscopic motions.
To summarize, the space of state variables is constituted by the quantities: ud , qd, and ub. qb. For future use,
we introduce also the diffusive and ballistic temperatures Td and Tb defined respectively by Td = ud /Cd, Tb
= ub / Cb where Cd and Cb have the dimension of a heat capacity per unit volume Although these
quantities bear some analogy with the classical definition of the temperature, it should however be realized
that, strictly speaking, these quantities do not represent temperatures in the usual sense but must be
considered as a measure of the internal energies
The next step consists in establishing the evolution equations for the set of basic variables. The evolutions
of the internal energies ud and ub are governed by balance laws of the form
u&
d = -.qd +rd ,
,
u&
b = -.qb +rb
(3)
while the total internal energy u  ud  ub satisfies the first law of thermodynamics
u& .q  r ,
(4)
the quantities rd , rb and r designate source terms which may be either positive or negative,in virtue of the
first law (4), one has to satisfy rd + rb = r; in absence of energy source ( r=0 ), one has simply that rd =rb. Based on kinetic theory considerations, it is shown that [1]
rb = -ub τ b ,
(5)
where the sign minus indicates that ballistic carriers can be converted into diffusive ones but that the
converse is not possible, τb is the relaxation time of the ballistic energy flux qb.
It remains to derive the evolution equation for the fluxes. Concerning the diffusive phonons, it is assumed
that they satisfy Cattaneo’s equation to cope with their high frequency properties and in agreement with the
general results of EIT; as a consequence, we are allowed to write
τ d q&
d + qd = - λd Td ,
(6)
wherein the relaxation time τd and the heat conductivity coefficient λd are positive quantities. However,
expression (6) is not able to describe the ballistic regime where most of the carriers cross the system without
experiencing collisions, except with the boundaries. To cope with this situation, it is necessary to introduce
non-locality. The simplest way to proceed is to complement (6) with additional terms involving  2 q and
.q so that the evolution equation of the ballistic heat flux will take the form
 b q&b + qb  bT  lb2 (2qb  2.qb ) ,
(7)
lb is related to the correlation length of the ballistic phonons which will be identified with the characteristic
length L of the system. Expression (7) has been justified within the framework of EIT [2] by selecting the
flux of the flux as extra variable besides q. Relation (7) is comparable to Guyer and Krumhansl’s equation
[2, 3] obtained from the phonon kinetic theory kinetic, the latter providing also a relation between heat
conductivity λ, relaxation time τ, heat capacity per unit volume Cv and sound velocity c0 of the form
1
2
  Cv c0  .
(8)
3
Expressions (3), (6) and (7) provide the basic set of eight scalar evolution equations for the eight
unknowns ud , ub ,qd and qb. After eliminating qd and qb and assuming that all the transport coefficients are
constant, one is led to the two second-order linear coupled differential equations :
2
&
&
&
τ d u&
d +ud - (τ d / τ b )ub - (1 / τ b )ub = (λd / Cd ) ud ,
2
2
&
&
&
τbu&
b + 2ub +ub / τb = (λb / Cb )∇ ub +3lb .[ub  (1/  b )ub ] .
(9)
(10)
Expressions (9) and (10) are the key relations of our model. In the particular case τd =τb =τ, making use of
(3b) and setting lb  0 , one recovers from expression (9) Chen’s result
2
&
&
τu&
d +ud + .qb = (λd / Cd ) ud ,
(11)
which differs from the telegraph equation that should be obtained from Cattaneo’ s law by the presence of
the term in . qb . In Chen’s formalism, the heat flux vector qb has been obtained by using the kinetic
definition of the heat flux and by solving Boltzmann’s equation.
Here, we will not refer to a kinetic approach and solve the problem exclusively at the macroscopic level. As
illustration, we apply the above considerations to the study of transient heat transport through a thin film
whose two faces are subject to a temperature difference. Numerical checking indicates that the results of
our model equations (12) and (13) are comparable to these of Chen.
It should however be underlined that our model is more general than Chen’s who introduced, without any
justification, the simplifying assumptions that τd = τb and Cd =Cb. Chen’s model is the result of a mixing
of macroscopic and microscopic approaches while our description is purely macroscopic. Finally, it should
be noticed that Chen remains silent about the signs of τi ,λi and C i (i= d,b).
4. APPLICATION: TRANSIENT HEAT TRANSPORT IN THIN FILMS
The foregoing model will be applied to the study of heat conduction in a one dimensional thin film of
thickness L smaller than the mean free path l of the phonons, heat capacity and heat conductivity are
assumed to be constant and to take the same values for the diffusive and ballistic phonons, internal energy
sources are absent (r=0). Initially, the system is at uniform temperature T0. The lower surface x=0 is
suddenly brought to temperature T1 = T0 + ΔT , while the upper surface x=L is kept at temperature T0.
Since the ballistic phonons originate at the walls, their energy is the prescribed energy at the boundaries,
i.e. Cv T0 and Cv T1 respectively. For further purpose, let us introduce the Knudsen number Kn = l /L;
since the mean free path of the ballistic carriers is comparable to the length of the material, one has
2
2
Knb  1 ,while for the diffusive phonons, it will be given the more general form ( Knd )  3 d / Cv L in
virtue of expression (8) and the result c0  ld /  d .
Having in mind numerical solutions, it is convenient to use dimensionless quantities
t*  t /  b , x*  x / L, b,d  (ub,d  CvT0 ) / Cv T ,   (u  2CvT0 ) / Cv T ,
(12)
with b ,  d and  ( b   d ) designating the non-dimensional temperature associated to the ballistic,
diffusive and total energy respectively. The corresponding evolution equations (9) and (10) take now the
form
Knd
 2 d ( Knb ) 2  2 d  d  d b
T



 b  0  0 ,
2
2
3 x *
t *  b t *
T
t *
2
2
 2b
b
T0 10
b
2  b
2 

2




(
Kn
)

3(
Kn
)
 0.
b
b
b
2
2
2
t *
T 3
t *
x *
x * t *
(13)
(14)
.
It remains to fix the initial and boundary conditions. Initially, the sample is at uniform temperature T0 (or
u(x,0) = CvT0) from which follows that the corresponding non-dimensional total energy is
b ( x*,0)  d ( x*,0)  T0 / T ,
(15)
As it is reasonable to admit that at short times, ballistic collisions are dominant so that the whole initial
energy if of ballistic nature with ub (x,0)=CvT0 and, in addition, assuming that the time derivatives of
both ballistic and diffusive energies are vanishing , we are allowed to write
t=0: b ( x,0)  0, d  T0 / T ,
(16)
b / t*  d / t*  0 .
(17)
As boundary conditions, we select
x*  0 : b (0, t*)  1,  d / x*  0,
x*  1: b (1, t*)  0,  d / x*  0,
(18)
this choice is motivated by the fact that ballistic collisions originates principally at the boundaries and that
the boundaries are supposed to be impervious to diffusive phonons.
Preliminary numerical analyses
indicate that the temperature profile is systematically lower than with Fourier’s model but a good
agreement with Chen’s and kinetic theory results is achieved.
5. CONCLUSIONS
A thermodynamic description of transient heat transport at nano-scales based on Extended Irreversible
Thermodynamics is proposed. The central assumption of the present work is that, contrary to previous
approaches, the set of variables - namely the internal energy and the energy flux - is split into contributions
of diffusive and ballistic nature. Heat transport is viewed as a two-fluid diffusion-reaction process with
ballistic particles converting into diffusive ones.
The corresponding time-evolution equations for the ensemble of variables have been formulated with a
special status reserved to ballistic carriers whose behaviour is mainly dictated by non-local effects. After
elimination of the heat fluxes, one has obtained two second order coupled differential equations for the
ballistic and diffusive internal energies which constitute the basic relations of our model. Our
thermodynamic phenomenological approach is inspired by a recent analysis [1] by Chen. This author has
developed a mixed macro-microscopic formalism wherein the diffusive regime is governed by Cattaneo’s
equation while ballistic particles are described by Boltzmann's equation. Chen's approach was recently
shown to be in good agreement with pure kinetic results. We are not only able to reproduce the main results
of Chen's ballistic-diffusion model but we have taken advantage of the thermodynamic approach to better
apprehend Chen's theory.
One of our objectives was to convince the reader of the flexibility and wide range of applicability of
Extended Irreversible Thermodynamics. It is shown that a rather simple model is able to cope with much of
the results derived from more sophisticated approaches.
It should nevertheless be realized that the formalism discussed above represents only a first step towards a
more elaborate description of heat transport at micro- and nano-scales. In particular, it must be noted that
the selection of state variables is not unique. Another choice, like the total and ballistic energies together
with their corresponding heat fluxes, is also allowed and would yield results similar to these of the present
work. It is also expected that higher order fluxes (the flux of the fluxes, the flux of the flux of the
fluxes,…) [5] should be introduced from the start to cope the particulate behaviour of heat carriers at short
wave lengths, but the difficulty is then the physical interpretation of the new variables coupled to the
complexity of the mathematical formalism. The selection of the most appropriate set of state variables
remains an open problem. In addition, for the sake of simplicity, all coupling terms have been omitted in
the Cattaneo and Guyer-Krumhansl relations (6) and (7), a more refined approach should include such
contributions. Moreover, the establishment of appropriate boundary conditions remains a delicate task.
From a more fundamental point of view, questions may also be raised about the definition of temperature
at nano-scales. To circumvent this problem in our analysis, temperature was understood as a measure of
internal energy.
All these considerations lead us naturally to the conclusion that heat transport in nano- and micro-systems
remains a challenging unfinished task.
ACKNOWLEDGEMENTS
The present study is supported by a project of collaboration between Wallonie-Bruxelles and Quebec
(biennale 2009-2011). Discussions with professors A. Palumbo (Messina University) and A. Valenti
(Catania University) about an earlier formulation of the model are also acknowledged.
REFERENCES
[1] G. Chen, Ballistic-diffusive equation for transient heat conduction from nano-to macroscales, ASME Jl. Heat Transfer 124
(2002) 320-331.
[2] D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics, 4rd edition, Springer, Berlin (2010).
[3] G .Lebon, D. Jou, J. Casas-Vazquez, Understanding Non-equilibrium Thermodynamics, Springer, Berlin (2008).
[4] C. Cattaneo, Sulla conduzione del calore, Atti del Seminario Matematico e Fisico della Università di Modena, 3 (1948) 3-21.
[5] X. Alvarez, D.Jou, Memory and nonlocal effects of heat transport: from diffusive to ballistic regimes, Applied Phys. Lett., 90
(2007) 083108.